[Remus nods, looking faintly pleased with himself.] Yes, really!

[The question has him chuckling, but he nods again, a little more enthusiastically.] Of course. Generally speaking, most witches and wizards are a little lost without their wands. There are some who can do wandless magic but that's limited greatly.

As for the hat...most wizards my age wouldn't be caught dead. It'd be like wearing a mortarboard around all the time.